In monetary economics, a money multiplier is one of various closely related ratios of commercial bank money to central bank money under a fractional-reserve banking system.^[1] Most often, it measures the maximum amount of commercial bank money that can be created by a given unit of central bank money. That is, in a fractional-reserve banking system, the total amount of loans that commercial banks are allowed to extend (the commercial bank money that they can legally create) is a multiple of reserves; this multiple is the reciprocal of the reserve ratio, and it is an economic multiplier.^[2]

Suppose some initial amount, d₁, is deposited into the banking system. With a reserve requirement of R, this deposit creates initial excess reserves equal to E₁ = (1 – R) x d₁. Assuming all of this amount is lent out and redeposited within the system, these excess reserves become new money: D M₁ = E₁ = (1 – R) x d₁. This second deposit creates its own excess reserves equal to E₂ = (1 – R) x D M₁ = (1 – R) x E₁. Again, this is redeposited as new money, so that D M₂ = (1 – R) x E₁. Continuing on like this indefinitely, we see a pattern develop:

D M₁ = E₁

D M₂ = E₂ = (1 – R) x E₁

D M₃ = E₃ = (1 – R) x E₂ = (1 – R)² x E₁

D M₄ = E₄ = (1 – R) x E₃ = (1 – R)³ x E₁

and so on ad infinitum.

The total increase in new money (call this “D“) can be found by adding up all the successive changes in new money, D = D M₁ + D M₂ + D M₃ + …. Then substituting for D M_i,
D =E₁ x [1 + (1 - R) + (1 - R)² + (1 - R)³ + ...].

Suppose we multiply D by the term (1 – R) and subtract this from D. All terms on the right-hand side with the exception of the initial “1″ would cancel out; the resulting difference is D – (1 – R) x D = E₁. Now collect the “D” terms on the left to obtain D x [1 - (1 - R)] = D x R = E₁. Finally, divide both sides by R to obtain the desired result: D = E₁ x . That is, an initial amount of excess reserves equal to E₁ creates new money equal to this amount multiplied by the inverse of the reserve requirement.

The deposit expansion multiplier

The easiest way to understand how the multiplier works is to consider what happens under simplifying assumptions:

• Banks keep a fixed fraction of deposits to meet the reserve requirement.
• Customers of the banks pay each other by cheque (or transfer etc.) but not by withdrawing cash to make payments.
• When customers do not receive these payments they do not withdraw any of the money from the bank.

Now consider a the following sequence of events

1. An initial deposit is made of £100
2. The bank is able to lend £90 of this. The borrower draws cheques against the £90 balance now in their account that the payees deposit in accounts in the same of other banks. Now customer balances have increased by the original £100 plus the £90 from the new cheque deposits: a total of £190
3. The bank can now lend 90% of the £90, a further £81.
4. Total deposits are now increased by another £81 to £ to £271
5. This process repeats and the total increase in bank deposits is 10 times the amount initially deposited: i.e. £1,000

As you might infer from the above, the multiplier is the reciprocal of the reserve requirement. If the reserve requirement was (a very high) 20% the multiplier would be 1 ÷ 0.2 = 5